Similar results hold for the configuration model. Take any degree sequence \((D_1,\ldots , D_n)\) , denoted by \(\mathbf {D}_n\) , the configuration model \( \mathbb {CM}(n,\mathbf {D}_n)\)  is constructed by uniformly randomly pairing \(D_1,\ldots ,D_n\)  half-edges attached to vertices \(1,\ldots , n\) , respectively. Two paired half-edges form a single edge. If \(D_i, 1\le i\le n\) , are i.i.d. sampled from a distribution \(D\)  (with minor adjustment to make sure that the sum \(\sum _{i=1}^n D_i\)  is even), then we may write \(\mathbb {CM}(n,D)\)  for \(\mathbb {CM}(n,\mathbf {D}_n)\) . It is known that the local neighborhood of \(\mathbb {CM}(n,D)\)  resembles a two-stage Galton-Watson tree where the offspring distribution of the root is \(D\)  while the offspring distribution of later generations is \(D^*-1\)  (\(D^*\)  is the size-biased version of \(D\) ).
See, e.g., [1]} for a detailed introduction to the configuration model.
Let \(z^{\prime }_0\)  be the survival probability for this two-stage Galton-Watson tree. Then for the SIR model on \(\mathbb {CM}(n,D)\) , \(\lambda _c=\gamma m_1/(m_2-2m_1)\)  where \(m_i\)  is the \(i\) -th moment of \(D\) . In addition, both the probability of an outbreak and the scaled final epidemic size (conditionally on an outbreak) converge to \(z^{\prime }_0\) , which goes to 0 as \(\lambda \rightarrow \lambda _c\) . See, e.g., [2]} for statements and proofs of these results.
